IL NUOVO CIMENTO

VOL. 102 A, N. 3

Settembre 1989

NOTE BREVI

The Reaction Rate at T-300 K for the Reactions 2H(p, ~,)3He, 2H(d, p) 3H, 2H(d, n) 3He, 3H(d, n) 4He, ~He(d, p) 4He, 2H(a, ~,)6Li, 3He(3He, 2p) 4He, 3He(a, ~,)7Be. A. SCALIA Dipartimento di Fisica dell'Universit4 - Catania Istituto Nazionale di Fisica Nucleare - Sezione di Catania

(ricevuto il 3 Maggio 1989)

Summary. - - The reaction rate is obtained for the reactions 2H(p, v)3He, 2H(d, p) 3H, 2H(d,n) 3He, 3H(d,n) 4He, 3He(d, p) 4He, 2H(a,~,)6Li, 3He(~He, 2p)4He, SHe(a,y)TBe at T - 3 0 0 K . The distribution of relative velocities between two different sets of particles is described by MaxwellBoltzmann distribution. The fusion cross-section is determined by using the extended elastic model II. PACS 25.70 - Heavy-ion-induced reactions and scattering.

In previous papers (~) we suggested for the thermonuclear reaction rate an expression obtained in the framework of the elastic model for the heavy-ion fusion (~). The values of the cross-section factor were obtained at temperature involved in nuclear astrophysics and they have been compared with those reported in the literature for several systems (1-4).

(~) A. SCALIA:Nuovo Cimento A, 109, 559 (1988). (2) A. SCALIA:Nuovo Ci?q~ento A, 101, 795 (1989). (8) A. SCALIA: The nuclear fusion for the reactions 2H(d, n)3He, 2H(d, p)~H at low energy, to be published in Nuovo Cimento A. (4) A. SCALIA:The nuclear fusion for the reaction 3H(d, n)4He at very low energy, to be published in Nuovo Cimento A. (5) A. SCALIA:Nuovo Cimento A, 98, 571 (1987). 953

954

A. SCALIA

In the present paper, by using the extended elastic model II (EEM II) (6.7), the reaction rate at T ~ 300 K is determined for the considered systems by assuming that the distribution of relative velocities between two different sets of particles will be described by Maxwell-Boltzmann distribution. Following the EEM II the fusion cross-section can be written as (1)

zf = ~f[1 - g(y)] = 5-f[1 - g(y)][1 - g, (y)] = ==

G(y)[1 + G(y)][1 - g(y)][1 - gl (Y)]

with

(2) g l ( y ) : e x p [ - ( l -d ~) Y~2. m789

,

y=ES-~sE,

d=E:,

k is the wave number, v is the Coulomb parameter, E is the centre~of-mass energy, EB and Es are two parameters expressed in MeV, which are determined by comparing the experimental values of fusion cross-section with -~f(1.5) (see fig. 1), ~ is the value of y at which 5-f attains the minimum value (see fig. 1), Ymis obtained by using 5-f in the expression of the cross-section factor (1), ~., and ~2 are determined by comparing the experimental values of fusion cross-section with zf(6,7). To determine Ym we consider the solution of the equation (3)

2 1 + 2G(y) d - y - M I (y) 1 + G(y) '

which satisfies the inequality (4)

Ym > 0.19 ;

M1 (y) is defined as

(5)

M1 (y) = [exp [exp [y]]] (exp [y]).

(6) A. SCALIA: The extended elastic model applied to the reaction SHe(a,v)TBe, to be published in Nuovo Cimento A. (~) A. SCALIA:The extended elastic model applied to the reaction 3He(~He,2p) 4He, to be published in Nuovo Cimento A.

THE

REACTION

RATE

AT

T~

300 K

FOR THE

REACTIONS

ETC.

955

We obtain y~ by solving eq. (1)

(6)

_

_

-{-

(d - y)2

_

M1 (y) ]2 1 + 2G(y) G(y) = M~ (y)[1 + f(y)] 1 + G(y) 1 + G(y) _

with

f(y) = exp [y].

(7)

The values of EB, Es, Ym, Em=EB--ymEs, Ym, Em=EB-ymEs, Y~, Y2 are reported in table I. A comparison between the experimental values of fusion cross-section(8) and those obtained by using eqs. (1)-(7) is shown in fig. 1 for the reaction 2H(p, T)3He, for the other reactions see ref. (2,3,6.~9.10). By using eqs. (1)-(7) the cross-section factor can be written as

(8)

d

with M(T) = (8=)1/2(Z1 Z2 e2)2 M1/2 [1~q~3/2 12\ ~xi ]

(9)

'

IT(Y)=d~yG(y)[l +G(y)]exp [

EB -- y E s rt

-~-~ ][l-g(y)][1-gl(y)],

M~2 is the reduced mass of particles i and 2, E is the centre-of-mass energy, K is Boltzmann's constant, T is the temperature. The solution in y of the equation (10)

d-y

+ Es ~=MI(Y)

1 + 2G(y) l+G(y)

g'(y)[1 - gl(Y)] + g{ [1 - g(y)] [1 - g(y)][1 - gl(Y)]

'

(8) G. M. GRIFFITHS and J. B. WARREN: Proc. Phys. Soc. A, 68, 781 (1955); G. M. GRIFFITHS, E. A. LARSON and L. P. ROBERTSON: Can. J. Phys., 40, 402 (1962); G. M. GRIFFITHS, M. LAL and C. D. SCARFE: Can. J. Phys., 41, 724 (1963). (9) A. SCALIA:The nuclearfusion for the reaction 3H(d, p)4He, to be published in Nuovo

Cimento A. (10) A. SCALIA: The extended elastic model II applied to the reaction 2H(~,~)~Li, submitted to Nuovo Cimento A. 61 - II Nuovo Cimento A.

956

A. SCALIA

10-3!

lO~

10-~

10-6

10 -7

,

I

200

,

I

,

&O0

I

,

I

600

800 Ec.m, (keY)

,

I

,

1000

I

,

1200

Fig. 1. - Comparison between the experimental values of fusion cross-section (8) and calculated ones. Full line zf, dashed line ~f.:f, ~f are defined in the text. with

g'(y) = -~yg(y),

(11)

gl(Y) -

gl(y),

gives the value ~ of y at which (12)

[ ~ y I ~ (y)]y= = 0.

In y = ~ it is (13)

dy

]y=~

so that I~ (y) attains in y = ~- a maximum value and the cross-section factor can be rewritten as

(14)

81/2

= M(T) I ~ ( ~ ) / ~ = (KT)~7~( - ~ e)~/i~(E) exp - ~

5 E,

6.085 1.128- 103 1.136.102 4.012.102 24.448 7 1.304 4 7.22.103

2H + 3He 2H + 4He 3He + 3He 3He + 4He 2H + 2H 2H + 3H 1H + 2H

T(K)

308.8 302.4 301.9 293.1 299.1 298.4 298.9 115.9

System

1H + 2H 2H + 2H 2H + 3H 2H + aHe 2H + 4He 3He + aHe 3He + 4He 2H + 2H

TABLE II.

E m (keV)

q

System

TABLE I.

0.053 0.13 0.13 0.48 0.031 0.27 0.15 0.05

E-(eV)

1.071 7 1.0354 0.867 1.030 7 1.764 5 1.085 35 8.448- 10 -1

Y-m

1.88 4.32 5.13 2.04 1.18 9.08 1.75 9.32

=e(b)

10-26

10-69 10-as

i0 -s5 10-139

10-24 10-21

10 -13

12.344 2.163.103 1.691 • 102 7.635.102 --1.041 • 103

E = (keV)

1.36 6.81 6.75 5.59 3.00 2.76 3.02" 6.69'

-E-/KT]

10-' 10 -3 10 -3 10 -9 10 -1 10 -~ 10 -3 10 -3

exp [ -

1.033 71 0.99641 0.829 0.991 7 --8.078

Ym

0.053 0.13 0.13 0.48 0.031 0.27 0.15 0.05

AE-(eV)

182.706 6 28.621 1 - 103 1.379.10 a 9.963- 10 a 196.483 9 41.778 8 8.00.103

EB (keV)

1.12.10 5.68.10 3.43.10 2.45.10

-31 -3° -53 -4

9 . 3 8 . 1 0 -110

3 . 1 9 . 1 0 -32 1 . 8 6 . 1 0 -43 2 . 0 0 . 1 0 -4°

( co >(cm a s- 1)

164.806 6 26.5536.10 a 1.460- l 0 s 9.277.103 225.029 7 37.291 7 8.62- 103

E s (keV)

20 2.5 10 1.8 --0.8

1"2

4.90' 10 -3

1.36" 10 -17

3.75- 10 69 4.48- 109 ? 1 . 1 3 . 1 0 -39

0 . 8 0 " 101

1.27.109 0.37- 10 2

r12 (cm-3 s-1)

9 ? 1.5 5 6 6 2.2

}'1

5O

z

©

>

>

>

t~

aa

958

A. SCALIA

with b E a suitable interval. The reaction r a t e can be obtained from eq. (6): (15)

r12 = (1 + ~12)-1 <~v> NIN2,

N1 and N2 are the n u m b e r densities of particles I and 2, respectively. In table I I are r e p o r t e d the values of E , z(E), e x p [ - E / K T ] , AE, <¢v>, r12 at T - 3 0 0 K . F o r AE we assume t h a t (16)

bE - E.

For N1 and N2 we assume t h a t (17)

N1 - 10 is cm -~ ,

N2 ~ 4.1022 cm -3 .

F r o m table I I it follows t h a t

(18)

r12 (1H + 2H) 1.27.109 r12 (2H + ~H) - 0 . 3 7 . 1 0 -2 - 3.43- 1011 .

W e note t h a t the value of reaction r a t e for the s y s t e m 2H + 4He is v e r y uncertain because the e x p e r i m e n t a l values of fusion cross-section known to the a u t h o r are relative to energies E > E m so t h a t r, cannot be d e t e r m i n e d (10).